Some challenges and results for causal and statistical inference with social network data

Transcription

1 slide 1 Some challenges and results for causal and statistical inference with social network data Elizabeth L. Ogburn Department of Biostatistics, Johns Hopkins University May 10, 2013

2 Network data evince complex forms of dependence among observations, making statistical inference difficult. Two challenges for causal inference using data sampled from a single social network: nonparametric identification of causal effects with interference, valid statistical inference under network dependence. slide 2

3 slide 3 Computer scientists, physicists and mathematicians have been researching networks for decades: topology, diffusion properties, generation,... very little statistics for outcomes on network nodes. Statisticians have been researching dependent data for decades: time series spatial data interference surprisingly little of this work is directly applicable to networks.

4 slide 4 outline Introduction and background Motivating example: HopeNet Design and aims. Briefly describe three distinct types of interference. Inference in the presence of network dependence. Explain why results for spatial dependence are not immediately applicable. Point towards some possible solutions to the problem of statistical inference in networks.

5 slide 5 background Interference requires a new framework and new tools for causal inference. (Cf. Rubin, 1990; Halloran & Struchiner, 1995; Sobel, 2006; Hong & Raudenbush, 2006; Vansteelandt, 2007; Rosenbaum, 2007; Hudgens & Halloran, 2008; Graham et al., 2010; Manski, 2010; Tchetgen Tchetgen & VanderWeele, 2012; Aronow& Samii, 2012; van der Laan, 2012) Existing methods for causal inference in the presence of interference have limitations in the social network context. Generally assume that treatment is randomized. Generally assume multiple independent groups are observed.

6 slide 6 background Recent availability of social network data has inspired methods and analyses, but with caveats: GEEs and GLMs have been used extensively but are usually inappropriate for network dependence. (Cf. Lyons, 2011; Shalizi & Thomas, 2011; VanderWeele, Ogburn & Tchetgen Tchetgen, 2012; Christakis & Fowler, 2012; Shalizi, 2012) Spatial autoregressive (SAR) parameterize an equilibrium state and generally lack causal interpretations. Recent work by van der Laan (2012) harnesses independence assumptions that are reasonable if dependence is due to contagion and network evolution is observed over time; other recent work relies on randomization-based inference (Cf. Bowers et al., 2013; Lunagomez & Airoldi, 2014; Rosenblum, 2007)

7 slide 7 HopeNet Study Health outcomes, progressive entrepreneurship, and Networks Nyakabare Parish, Mbarara Province, SW Uganda 8villages 2000 adults

8 slide 8 HopeNet Study Complete social network census for the adult population of the parish. Clean water intervention. Microenterprise intervention.

9 effects of interest What types of effects are we interested in? Under what conditions are these effects nonparametrically identifiable? Ogburn & VanderWeele (2012) Under what conditions can we perform inference about them? slide 9

10 slide 10 definitions Stationarity: features of the distribution of observations does not depend on location in the network. Distance i,j between nodes i and j: lengthofshortestpath connecting i and j. Other definitions are possible, e.g. taking into account number of paths or average path length between i and j. M-dependence: W i W j if i,j > m. Mixing conditions: Cov (W i,w j ) 0as i,j.

11 slide 11 inference with network dependence Sources of network dependence. Limitations of methods for spatial dependence. Some methods for statistical inference in networks. local dependence weakly dependent clusters subsampling k-dependence

12 slide 12 sources of network dependence Latent variables cause outcomes among close social contacts to be more correlated than among distant contacts. (E.g. homophily, geography, shared culture.)

13 sources of network dependence Contagion implies information barrier structures e.g. [ Y1 t Y 2 t Y 1 t 2, Y2 t 2, Y1 t 1, and Y2 t 1 ] [ and Y t 2 1 Y3 t 1 ]. When a network is observed at a single time point, this will resemble latent variable dependence. If network is observed before information has had the opportunity to diffuse, m-dependence would be reasonable. slide 13

14 slide 14 Why can t we use spatial dependence results? Network topology doesn t naturally correspond to Euclidean space. In order to embed a network in R d,wewouldhavetoletd grow with sample size n. Spatial results require d to be fixed or to grow slowly with n. Population growth is usually assumed to occur at the boundaries of the d-dimensional space. It s not clear how to define boundaries in networks (nor how to define population growth). Mixing assumptions and m-dependence don t imply bounded correlation structure. In spatial data most observations are distant from one another. The maximum network-based distance between two observations may be very small. The distance distribution may not be right-skewed enough.

15 slide 15 inference with network dependence Focusing for now on a traditional frequentist inferential framework, point estimates are generally unbiased; the challenge is for s.e.and interval estimation. Estimates of s.e. may be unbiased but not consistent.

16 slide 16 inference with network dependence For simplicity, suppose our target parameter is a population mean, µ. The sample mean, W = n i=1 W i,isunbiasedforµ. The problem is consistent estimation of the asymptotic variance Avar ( W ). Agnostic about sources of dependence (contagion vs. latent variables). Population growth must preserve key features of the network and data.

17 slide 17 local dependence Local dependence: for each observation W i,thereisasetofindicesi such that W i W j for j I c. M-dependence is a special case. Using Stein s method, it is easy to show that CLTs hold for locally dependent data, under restrictions on the size of I. (Cf. Chen, 1978; Barbour, Karonski & Rucinski, 1989; Rinott & Rotar, 1996; Raic, 2002; Chen & Shao, 2005) What types of networks are consistent with the restrictions on I?

18 slide 18 weakly dependent clusters If K clusters are asymptotically mean independent from one another, there are two approaches we might consider: 1. T distribution based confidence intervals (Ibragimov & Muller, 2010; Bester, Conley & Hansen, 2011). Requires asymptotic normality and mean stationarity at the cluster level. 2. Bootstrap the weakly dependent communities. Stationarity is required only at the cluster level. In the spatial dependence literature mean independence is justified with conditions on the relative size of the boundaries and interiors of the clusters; growth in d dimensions uniformly. These conditions don t translate into the network setting...

19 slide 19 subsampling Subsampling has been used in many spatial dependence contexts (cf. Lahiri, 2003; Politis, Romano & Wolf, 1999), butneithertheimplementation nor the conditions under which it is appropriate are immediately applicable to networks. Under mild stationarity and dependence conditions, we can subsample to estimate Avar ( W ): 1. Select B subsamples of consecutive observations. 2. In each subsample, calculate the subsample variance estimator σ 2 b. 3. Estimate Avar( W ) with the average of the subsample estimators: σ 2 W = 1 B B b=1 σ 2 b.

20 slide 20 subsampling This is reasonable if, as n b and n, 1. σ 2 b is asymptotically unbiased for Avar( W ). This will hold if key features of the network and the mean and variance of W are stationary over groups smaller than the subsample sizes. 2. Var ( σ 2 W ) 0, where ) 1 Var ( σ 2 W = B 2 B Var ( σ 2 ) b b=1 + 1 B 2 2Cov ( σ b 2, σ 2 ) d I b,i d m + 1 B 2 2Cov ( σ b 2, σ 2 ) d I b,i d >m (1) (2) (3)

21 slide 21 k-dependence ) In some settings it may be expedient to estimate Cov (W directly. K-dependence: Cov(W i,w j )=σ k,wherek = i,j. Under k-dependence, m-dependence, and mean stationarity, ) we can get an unbiased and consistent estimate of Cov (W by this procedure: 1. For each k < m, selectpairsofnodesthatarek units apart, such that the pairs themselves are at least m units apart from on another. 2. Estimate σ k with ) the average covariance across the selected pairs. 3. Estimate Cov (W with the plug-in estimator. This doesn t demand as much from m-dependence as other procedures do...

22 slide 22 other directions Different types of asymptotics: combine infill and increasing domain asymptotics, fractal asymptotics. Identify low-level conditions for CLTs and LLNs in network settings. Learn a new, latent distance metric. (E.g. work by Adrian Raftery & colleagues) Finite sample results using bounded influence: Var (W i ) >> j Cov(W i,w j ).

23 slide 23 Thank you

24 slide 24 references Rubin, D. B. (1990). On the application of probability theory toagriculturalexperiments. essay on principles. section 9. comment: Neyman (1923) and causal inference in experiments and observational studies. Statistical Science 5, Halloran, M. E. & Struchiner, C. J. (1995). Causal inference in infectious diseases. Epidemiology, Sobel, M. E. (2006). What do randomized studies of housing mobility demonstrate? Journal of the American Statistical Association 101, Hong, G. & Raudenbush, S. W. (2008). Causal inference for time-varying instruc- tional treatments. Journal of Educational and Behavioral Statistics 33, Vansteelandt, S. (2007). On confounding, prediction and efficiency in the analysis of longitudinal and cross-sectional clustered data. Scandinavian Journal of Statistics 34, Rosenbaum, P. R. (2007). Interference between units in randomized experiments. Journal of the American Statistical Association 102, Hudgens, M. G. & Halloran, M. E. (2008). Toward causal inference with inter- ference. Journal of the American Statistical Association 103,

25 slide 25 references Graham, B. S., Imbens, G. W. & Ridder, G. (2010). Measuring the effectsof segregation in the presence of social spillovers: A nonparametric approach. Tech. rep., National Bureau of Economic Research. Manski, C. F. (In press). Identification of treatment response with social interactions. The Econometrics Journal. Tchetgen Tchetgen, E. J. & VanderWeele, T. J. (2012). On causal inference in the presence of interference. Statistical Methods in Medical Research 21, Aronow, P. & Samii, C. (2012). Estimating Causal EffectsUnder General Interference. Working paper ( van der Laan, M. J. (2012). Causal Inference for Networks. U.C. Berkeley Division of Biostatistics Working Paper Series Paper 300. Lyons, R. (2011). The spread of evidence-poor medicine via áawed social- network analyses. Statistics, Politics and Policy 2(1), Article 2, VanderWeele, T.J., Ogburn, E.L. & Tchetgen Tchetgen, E.J. (2012). Why and When "Flawed" Social Network Analyses Still Yield Valid Tests of no Contagion. Statistics, Politics, and Policy 3(1).

26 slide 26 references Christakis, N.A. and Fowler, J.H. (2011). Social contagion theory: examining dynamic social networks and human behavior. Statistics in Medicine. Shalizi, C.R., Thomas, A.C. (2011). Homophily and contagion aregenerically confounded in observational social network studies. Sociological Methods and Research, 40: Shalizi C.R. (2012). Comment on Why and When Flawed Social Network Analyses Still Yield Valid Tests of no Contagion. Statistics, Politics, and Policy 3(1). Ogburn, E.L. & VanderWeele, T.J. (2012). Causal diagrams for interferenceand contagion. Under revision. Chen, L.H.Y. (1978). Two central limit problems for dependent random variables. Probability Theory and Related Fields 43, no. 3: Barbour, A. D., Karoński, M., & Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B, 47(2), Rinott, Y., & Rotar, V. (1996). A Multivariate CLT for Local Dependence with n 1/2 log n Rate and Applications to Multivariate Graph Related Statistics. Journal of multivariate analysis, 56,

27 slide 27 references Raic, M. (2003). Normal approximation by Stein s method. In Proceedings of the seventh young statisticians meeting. Chen, L. H., & Shao, Q. M. (2005). Stein s method for normal approximation. An introduction to Stein s method, 4, Chen, L. H., & Shao, Q. M. (2004). Normal approximation under local dependence. The Annals of Probability, 32(3), Ibragimov, R., & Müller, U. K. (2010). T-statistic based correlation and heterogeneity robust inference. Journal of Business & Economic Statistics, 28(4), Bester, C. A., Conley, T. G., & Hansen, C. B. (2011). Inference withdependentdata using cluster covariance estimators. Journal of Econometrics, 165(2), Lahiri, S. N. (2003). Resampling methods for dependent data. Springer. Politis, D., Romano, J. P., & Wolf, M. (1999). Weak convergence of dependent empirical measures with application to subsampling in function spaces. Journal of statistical planning and inference, 79(2),

28 slide 28 sources of network dependence 1. Latent variables cause outcomes among close social contacts to be more correlated than among distant contacts. (E.g. homophily, geography, shared culture.) This is the type of dependence considered in the context of spatial data. 2. Direct interference implies densely overlapping clustering of exposure status (cf. Aronow & Samii, 2012). Outcomes are independent conditional on exposure, unless other forms of dependence are also present.

29 When a network is observed at a single time point, this will resemble latent variable dependence. If network is observed before information has had the opportunity to diffuse, m-dependence would be reasonable. slide 29 sources of network dependence 3. Contagion implies an information barrier structure: [ Y t 1 Y2 t Y 1 t 2, Y2 t 2, Y1 t 1, and Y2 t 1 ] [ and Y t 2 1 Y3 t 1 ].